TITLE: Plot of Ramanujan tau function
QUESTION [10 upvotes]: There is a picture on wikipedia of Ramanujan tau function. At first I noticed that there are exceptional red point (where the red points are sparse in the lower part), this should be due to Sato-Tate conjecture. 
But it seems strange since where red points are dense blue points  are sparse!
Is there any explanation about these? 
(Wiki: The blue line picks only the values of n that are multiples of 121.)

By the way, is there any conjecture about the growth （a lower bound） of this function? (Surely stronger than the Lehmer's conjecture.)

REPLY [3 votes]: The best lower bound known seems to be due to Ram Murty:
$$\tau(n)=\Omega (n^{11/2}e^{c \log n / \log \log n})$$
for some $c>0$ absolute and effective (note: thanks to the Sato-Tate conjecture might be able to take $c<\log 2$).
This result is proved in:

Ram Murty, Some Omega results for Ramanujan's tau function (1982)

In their book on Ramanujan, Ram and Kummar mention:

This result is essentially best possible since we know that
$$d(n)<e^{c' \log n / \log \log n}$$

It might be worth mention that this holds for arbitrary cusp forms, for some $c>0$,
$$a(n)=\Omega (n^{(k-1)/2}e^{c \log n / \log \log n})$$
The exceptional values you observe seem to be related to congruences modulo $11$ and $121$, but I have no idea about that. Ramanujan himself certainly had much to say on the matter:

Srinivasa Ramanujan, Properties of $p(n)$ and $\tau (n)$... (unpublished)

Hopefully someone can answer that part of your question better.